limit inferior
The limit inferior is the mirror image of the limit superior: the 'eventual low-water mark' of a sequence. It is the smallest value the terms keep approaching infinitely often — the floor they repeatedly sink to (or toward), no matter how high they may keep bouncing back up.
As with limsup there are two equivalent descriptions. First, liminf a_n is the smallest subsequential limit (allowing -infinity). Second, for computation: liminf a_n = lim_{n->inf} ( inf_{k >= n} a_k ). The inner infimum inf_{k >= n} a_k is the lowest value in the tail from index n onward; as the tail shrinks these infima increase, and their limit is the liminf.
Like the limsup, the liminf exists for every sequence in the extended reals, and one always has liminf a_n <= limsup a_n. Equality of the two, to a common finite value L, is exactly the condition for a_n -> L. The liminf and limsup thus bracket all the long-run behavior of a sequence between a guaranteed floor and a guaranteed ceiling.
For a_n = (-1)^n, liminf a_n = -1 and limsup a_n = 1. For the convergent a_n = 1/n, liminf and limsup both equal 0, consistent with 1/n -> 0.
liminf = limsup is a clean test for convergence, valid for every sequence.